Golden Ratio Anyons from SU(2)_3 Chern-Simons Theory for Quantum Computing — E8 Intelligence Research

FINDING: SU(2)_3 Chern-Simons theory yields non-Abelian anyons with quantum dimension equal to the golden ratio φ = (1+√5)/2, a key resource for topological quantum computing. MATH: - SU(2)_k Chern-Simons level k=3. - Quantum dimension of the non-Abelian anyon (spin-1/2 representation): \( d = \frac{\sin(3\pi/5)}{\sin(\pi/5)} = \frac{\sin(108^\circ)}{\sin(36^\circ)} = \frac{\sqrt{10+2\sqrt{5}}/4}{\sqrt{10-2\sqrt{5}}/4} = \frac{1+\sqrt{5}}{2} = \varphi \approx 1.618 \). - Fusion rule: \( \phi \times \phi = 1 + \phi \) (Fibonacci anyon). - Central charge: \( c = \frac{3k}{k+2} = \frac{9}{5} = 1.8 \). - Modular S-matrix entry: \( S_{12} = \sqrt{\frac{2}{k+2}} \sin\left(\frac{2\pi}{k+2}\right) = \sqrt{\frac{2}{5}} \sin\left(\frac{2\pi}{5}\right) = \sqrt{\frac{2}{5}} \cdot \frac{\sqrt{10+2\sqrt{5}}}{4} \), related to φ. CONNECTION: - Golden ratio φ = 1.618 appears directly as the quantum dimension, linking topological order to pentagonal symmetry (root system H₂, icosahedral g Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com